The Hermite-Hadamard inequality revisited: Some new proofs and applications

New proofs of the classical Hermite-Hadamard inequality are presented and several applications are given, including Hadamard-type inequalities for the functions, whose derivatives have inflection points or whose derivatives are convex. Morever, some estimates from below and above for the first moments of functions $$ about the center point $c=(a+b)/2$ are obtained and the reverse Hardy inequality for convex functions $f:[0,\infty )\rightarrow (0,\infty )$ is established.